In 1770, Waring proposed the study of representing an integer as a sum of s perfect k-th powers. Over the past century, the Hardy-Littlewood circle method has been honed to produce an asymptotic for the number of such representations, with a central goal being to reduce the number of variables required. In the Hardy-Littlewood strategy, a critical step is to estimate a relevant exponential sum, which for the past seventy years has been approached via increasingly sophisticated versions of Vinogradov's mean value method. In recent years, Wooley has pushed the field ever closer to a final resolution of the main conjecture, called the Vinogradov Mean Value Theorem, via his efficient congruencing method. Now, by approaching the problem from the perspective of $l^2$ decoupling, Bourgain, Demeter and Guth have finally resolved the main conjecture. This lecture will survey these two approaches to the Vinogradov Mean Value Theorem, and several consequences for discrete restriction problems, Waring's problem, and the Riemann zeta function.

[After Bourgain, Demeter and Guth, and Wooley]